Boolean Laws and Teorems - Go by the rule

Sorry, could not post the notes as scheduled, was little busy with other thing. Today giving some Boolean Laws and Theorems. Boolean logic was developed by an English mathematician George Boole, used to construct and solve Boolean algebra and helped to design the electrical circuits. The basic laws of Boolean algebra:
Proof (Using Truth Table)
A + A.B = A						A( A + B) = A
A	B	A.B	A+A.B				A		B	A+B	A.(A+B
0	0	0	0				0		0	0	0
0	1	0	0				0		1	1	0
1	0	0	1				1		0	1	1
1	1	1	1				1		1	1	1
Proof (Using Postulate)
If A and B  are elements of a Boolean algebra
A+A.B = A								A( A + B) = A
A.1+A.B				A.1 = A				A.A + A.B		Distributive Law
A(1 + B)				Distributive Law				A + A.B		Idempotent law
A.1				Law on the property of 1				A.1 + A.B		A.1 = A
A				A.1 = A				A(1 +  B)		A.1
								A		A
Now just tell you about principle of duality. Principle of Duality The dual of any statement in a Boolean algebra is the statement obtained by interchanging OR with AND, and simultaneously inter-changing the elements 0 and 1 in the statement.

IdentityBoolean								Dual
A + 0 = A								A.1 = A
A + 1 = 1								A.0 = 0
A + A = A								A.A=A
A + A’ = 1								A.A’ = 0
A + B = B + A								A . B = B . A
A+ B.C = (A + B)(A + C)								A(B + C) = A . B + A. C
A( A + B) = A								A+A.B=A
A + AB = A + B								A.(A+B) = A
(A+ B)’ = A’.B’								(AB)’=A’ + B’
DeMorgan’s theorems provide mathematical verification of: : 1. The equivalency of the NAND and negative-OR gates 2. The equivalency of the NOR and negative-AND gates.
(A.B)'    =		A'+B'						(A + B)'       =			A'.B'
**Change the sign and break the line.
(A.B)' = A'+B'								(A + B)' = A'.B'
A	B	A.B	(A.B)'	A'	B'	A'+B'		A	B	A+B	(A+B)'	A'	B'	A'.B'
0	0	0	1	1	1	1		0	0	0	1	1	1	1
0	1	0	1	1	0	1		0	1	1	0	1	0	0
1	0	0	1	0	1	1		1	0	1	0	0	1	0
1	1	1	0	0	0	0		1	1	1	0	0	0	0
Next time it will be time for Logic circuits using Logic Gates.
!!!Wish all of you a Shuvo Vijaya Dashami / Happy Dussera!!!